Hi and welcome to the forum.
"Advanced writing skills" is a very wide topic, so what to cover will depend on a few things:
- Has your employer given you any guidance in terms of the genres of writing they want you to cover, or have they left it completely open?
- How many students do you have and how long is the course?
- If the content of the course has been left open, then the first thing you'll need to do is find out from your students what they need to write, or are interested in writing, in English. There are many different genres of writing, involving diferent skills and therefore different approaches, so if you can narrow it down, so much the better. For example, if your students are all businesspeople who need to write professional emails, then it is no good spending time working on how to write an academic essay, and vice versa. Perhaps they are journalism students and would like to develop skills in writing magazine articles. And so on...
So in the first class, I would spend some time getting your students talking about the kind of writing they need/want to do. Feel free to get back to me in the forum when you know a little more, and I'll be able to give you some more specific pointers about how to approach teaching these skills.
It would also be a good idea during the first class to get a sample of the students' writing, so you can see their starting point. Let's say, for example, that they are students and need to work on academic essays. While I wouldn't recommend getting them to write an essay during the class, it is something you could set for homework - give them a title and ask them for an essay of, say, 300 words. This will give you an idea of their individual strengths and weaknesses, as well as the different skills that they're lacking in this genre. It wil also give you useful material to use as the course progresses (having them rewrite this original essay using a skill that you have worked on during one of the classes, for example).
Setting the actual writing tasks for homework rather than during class time is a good idea generally - the class time should be used working on developing the skills that they'll need for these tasks. As I say, let me know when you have some concrete ideas of the type of writing they need, and I'll give you some pointers in terms of the types of skill to work on, and how to go about this.
In the first lesson, if you need a fun writing warm-up activity, try the one below - it's good for group dynamics and bonding, and can make the students feel good about writing - it makes it fun, which can help to counter the often negative student reaction to writing:
1. Give each student a pile of small pieces of paper.
2. Explain that to get to know each other, they can write questions to each other on these pieces of paper. Leave it completely open (although you may need to set content rules depending on the cultural context in which you are teaching - for you to decide what is appropriate here)
3. Explain that you will be the postman. When they have written their question, they should indicate on the paper the name of the sender (themselves) and who the question is addressed to. They should then call out "Postman" (or whatever you like) at which point you will come and collect their piece of paper and deliver it to the named recipient.
4. The recipient can reply or not, as they choose.
5. As this gets going and the students warm up, you will have cries of "Postman" coming thick and fast, and you will be running around the classroom tryin to keep up with all the deliveries.
6. Stop the activity after a suitable time (when delivery dies down or they look like they've had enough).
7. If the students agree, you can collect some of these written conversations at the end and this will give you a further sample of their writing.
Hope that helps.